| | Category | MA | P25 | Converging logarithmic spirals on the complex plane |
| | Abstract | The logarithm of a complex number in the form a+bi can be complicated to |
| | find and is volatile when graphed on the complex plane; however, |
| | continuously taking the logarithm of a complex number and taking the |
| | logarithm of the answer eventually gives a unique spiral that always |
| | converges to the same constant value. I started my project during the |
| | summer of 2009 with the help of Professor Jerold Grossman from Oakland |
| | University. He suggested many books and websites for me to use in order |
| | to research my problem. Since operations with complex numbers get |
| | really complex, I bought a mathematics program called Maple 13(used by |
| | many universities and college students). With the help of the program, I |
| | gathered data as proof of my findings. I created tables where I showed |
| | the patterns of my mathematical operations and used Maple to create plots |
| | that visually showed these numbers. To further my research, I went on to |
| | prove that my function does converge for many, if not all, values in the |
| | form a+bi. My proof includes the use of the Cauchy’s convergence |
| | theorem along with some unique proofs of my own. |
| | Bibliography | http://math.fullerton.edu/mathews/c2003/ComplexFunLogarithmMod.html |
| | |
| | http://mathworld.wolfram.com/Logarithm.html |
| | |
| | http://www.geom.uiuc.edu/~banchoff/script/CFGExp.html |